Before analyzing the PGROT, we introduce the following lemma.
 [1]} 
Let \(x^* \in \mathbb {R}^n\)  be the solution to the problem (REF ) and \(\eta = y-Ax^*\)  be the error.
Denote \(S = \operatorname{supp}(x^*)\)  and \(S^{p+1} = \operatorname{supp}(x^{p+1})\) .
Let \(u^p\)  and \(w^p\)  be the vector defined as in PGROT, and \(w^* \in \lbrace 0,1\rbrace ^n\)  be a binary \(k\) -sparse vector such that \(S \subseteq \operatorname{supp}(w^*)\) .
Then

(x*-up wp)S Sp+12  1+k1-2 k (x*-up) w*2 + 21-2 k2
